Find Slope of Tangent Line at P(1,2): Answer = 5/2

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In summary, the conversation is about finding the slope of the tangent line to a given curve at a given point. The speaker shares their approach and confusion in solving the problem, as well as their realization and clarification about the concept of square roots as a function. The expert summarizer also clarifies the difference between a function and a relation.
  • #1
mmapcpro
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Hi,

I was looking at a problem from an old test and I got confused about something.

Q. Find the slope of the tangent line to the given curve at a given point:

y = x(x^2 + 3)^1/2 ; P(1,2)

S. First I found the derivative:
[(x^2 + 3)^1/2] + [(x^2)(x^2 + 3)^-1/2]

Plugged in the values for x and y at the given point:
[(4)^1/2] + [(4)^-1/2]

The choices for answers were:
a) 2 b) -5/2 c) 3/2 d) 5/2 e) none of these

Square root of 4 can be + OR - 2, no?

That would mean I can have 4 possible solutions:
(4/2) + (1/2) = 5/2
(4/2) + (-1/2) = 3/2
(-4/2) + (1/2) = -3/2
(-4/2) + (-1/2) = -5/2

Since 3 of these are listed as choices, how did I pick the right one? (I chose d)5/2 and it was marked correct)

Forgive me if this is shockingly stupid...I am trying to review all my calculus and physics to start classes up again in the winter.
 
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  • #2
NEVERMIND!

I feel dumb...

Of course, the original equation must also hold true, which means that (4)^1/2 can only be +2
 
  • #3
In addition, "square root" is a FUNCTION which means it can have only one value. The square root of any positive number, a, is, by definition, the POSITIVE number, x, such that x2= a.

Of course, x2= a has TWO solutions: they are [sqrt](a) and -[sqrt](a).
 
  • #4
Originally posted by HallsofIvy
In addition, "square root" is a FUNCTION which means it can have only one value. The square root of any positive number, a, is, by definition, the POSITIVE number, x, such that x2= a.

Of course, x2= a has TWO solutions: they are [sqrt](a) and -[sqrt](a).


When you say FUNCTION, do you actually mean CONTINOUS ONE-TO-ONE FUNCTION?
 
  • #5
When I say FUNCTION, believe it or not, I mean FUNCTION. I do not meant "continuous" since that is not part of the definition of FUNCTION, although, in this problem, since the original post assumed the function was differentiable, it must be continuous. Nor do I mean "one-to-one". I consider "f(x)= x<sup>2</sup>", which is NOT one-to-one (f(2)= f(-2)), a perfectly good function. I do require that functions on the real numbers be "well-defined"- that is, that a single value of x can give only one value of f(x). That is, after all, the distinction between a "function" and "relation".
 

1. What does "Find Slope of Tangent Line" mean?

The slope of a tangent line at a given point on a curve represents the rate of change of the curve at that point. It tells us how steep the curve is at that specific point.

2. What does "P(1,2)" refer to?

P(1,2) represents the coordinates of the given point on the curve, with 1 being the x-coordinate and 2 being the y-coordinate. In this case, the point is located at x=1 and y=2.

3. How is the slope of a tangent line calculated?

The slope of a tangent line is calculated using the derivative of the function at the given point. In this case, the derivative would be evaluated at x=1 to find the slope of the tangent line at that point.

4. What is the significance of finding the slope of a tangent line?

The slope of a tangent line helps us understand the behavior of a curve at a specific point. It can also be used to find the equation of the tangent line, which can be useful in applications such as optimization and curve sketching.

5. What is the answer to "Find Slope of Tangent Line at P(1,2)"?

The answer is 5/2, which represents the slope of the tangent line at the point P(1,2) on the given curve. It tells us that the curve is increasing at a rate of 5/2 units for every 1 unit increase in the x-direction at x=1.

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